BCS to BEC Crossover and the Unitarity Fermi Gas. Mohit Randeria Ohio State University Boulder School on Modern aspects of Superconductivity

Transcription

1 1 BCS to BEC Crossover and the Unitarity Fermi Gas Mohit Randeria Ohio State University 2014 Boulder School on Modern aspects of Superconductivity

2 Review articles: 2 M. Randeria and E. Taylor, Ann. Rev. Cond. Mat. Phys. 5, 209 (2014) Introductory review by M. Randeria, W. Zwerger & M. Zwierlein, Plus 13 Chapters by leading experts in: BCS-BEC Crossover and the Unitary Fermi Gas ed. by W. Zwerger (Springer, 2012) S. Giorgini, L. P. Pitaevskii and S. Stringari, Rev. Mod. Phys. 80, 1215 (2008). Ultracold Fermi Gases, Proc. of the Varenna Enrico Fermi Summer School 2006, W. Ketterle, M. Inguscio and C. Salomon (editors).

3 Outline: (pre)history & Introduction Qualitative ideas of BCS-BEC crossover Theoretical progress Some key experiments Exact Results for strongly interacting regime Connections to other areas in physics Outlook

4 BCS ( ) o Fermions o Pairing o Condensation of pairs o superconductivity BEC ( ) o Bosons o Macroscopic occupation of a quantum state o superfluidity

5 Before 2004: BCS-BEC crossover was a problem of purely theoretical interest with diverse motivations, but no direct experimental relevance! o D. M. Eagles (1969) [Doped semiconductor SrTiO 3 ] o A. J. Leggett (1980) [Superfluid He 3 ] o P. Nozieres & S. Schmitt-Rink (1985) [Excitons, ] o M. Randeria & collaborators (1990 s) [HTSC]

6 2004: BCS-BEC crossover in ultracold Fermi gases Realized in the lab! Experiments: Jin (JILA) Ketterle (MIT) Salomon (ENS) Grimm (Innsbruck) Hulet(Rice) Thomas (Duke) 6 Li, 40 K atoms dilute : k 1 F 0.3µm E F 100 nk 1 mk spin up & down two hyperfine states [Jin group] Tunable two-body interaction: Feshbach Resonance a!1

7 7 Feshbach Resonance: Two-channel description ß à Single channel model* *broad resonance Tuning parameter = B-field Cheng, Grimm, Julienne & Tiesinga. RMP (2010) See: Leo Radizihovsky s lectures (k F r 0 1)

8 Feshbach Resonance (simplified) 8 Two-body problem in 3D: kr 0 1 Low-energy effective interaction: s-wave scattering length a d d = f 2 Unitarity à a = 1 threshold for à two-body bound state (in vacuum)

9 Attractive Fermi Gas: Dilute Gas: range o o g( )! a 1 g( ) = m 4 a r 0 k 1 F interparticle distance Dimensionless Coupling constant X k< m ' 1!1 k 2 r 0 For an equivalent real-space approach with range See: Y. Castin & F. Werner in Zwerger Book r 0! 0 9

10 BCS-BEC crossover [Leggett (1980); Nozieres & Schmitt-Rink (1985)] 10 V (r) V (r) V (r) a<0 a = 1 a>0 No bound state 1/k F a Threshold bound state BCS limit cooperative Cooper pairing pair size Unitarity a = 1 pair size strongly interacting gas BEC limit tightly bound molecules pair size

11 Outline: (pre)history & Introduction Qualitative ideas of BCS-BEC crossover Theoretical progress Some key experiments Exact Results for strongly interacting regime Connections to other areas in physics Outlook

12 Qualitative description of BCS-BEC crossover Based on: Sa demelo, MR & Engelbrecht, PRL (1993) 12

13 T=0 BCS-Leggett Mean Field Theory: 13 SadeMelo, MR, Engelbrecht, PRL (93), PRB(97) MFT Qualitatively correct at T=0: all the way from Cooper pairs to composite bosons! will address Quantitative limitations later Note: crossover region No small parameter near unitarity!

14 Energy Gap for Fermionic Excitations: BCS regime ( weak pairing ) 14 BEC regime ( strong pairing ) à Phase transition (not a crossover) for non-s-wave pairing, e.g. p+ip [Read & Green] Gapless Goldstone excitations: phonon BCS BEC Petrov, Salomon, Shlyapnikov a b =0.6a

15 Finite Temperatures T*: Pairing saddle-point/mft Saha ionization maximum T c E f ' 0.2 BCS limit Tc: Phase Coherence saddle-point + Gaussian fluctuations = NSR G-MB correction SadeMelo, Randeria & Engelbrecht, PRL (1993)

16 Evolution from Normal Fermi à Normal Bose Gas? Is the system quantum degenerate at these high T? *Pairing Pseudogap Randeria, Trivedi, Scalettar & Moreo PRL (1992) Trivedi & Randeria, PRL (1995) à possible Breakdown of Fermi-liquid description in pseudogap regime Recent QMC: Tc = 0.15 Ef Burovski et al, PRL (2008) T* = 0.2 Ef Magierski et al, PRL (2009) Experiments? See later!

17 Outline: Outlook (pre)history & Introduction Qualitative ideas of BCS-BEC crossover Theoretical progress No small parameter at unitarity! * Field theory * Quantum Monte Carlo Some key experiments Exact Results for strongly interacting regime Connections to other areas in physics

18 Unitary Fermi Gas 18 * MFT & NSR: ε-expansion: [26] SadeMelo, Randeria & Engelbrecht. PRL 71, 3202 (1993) [27] Engelbrecht, Randeria, & SadeMelo. PRB 55, (1997) [51] Nishida & Son, PRA (2007) and in The BCS-BEC Crossover and the Unitary Fermi Gas, ed. W. Zwerger, (Springer, 2011) QMC: Experiments: [57] Astrakharchik et al., PRL 93, (2004) [58] Bulgac, Drut & Magierski. PRA 78, (2008) [60] Carlson & Reddy. PRL 95, (2005) [62] Burovski et al, PRL 96,160402, (2006) [63] Burovski et al, PRL 101, (2008) [61] Schirotzek et al, PRL 101, (2008) [65] Ku et al, Science 335, 563 (2012) *revised [G. Zurn, PRL 110, (2013)] Table from: Randeria & Taylor Ann. Rev. CMP (2014)

19 Universality: Results, independent of microscopic details (e.g. Li, K, ) across the entire BCS-BEC crossover provided k F r All (dimensionless) results can be expressed as F(T/E f, 1/k F a) No interaction scale at Unitarity a = 1 Universal results for All observables F(T/E F ) Bertsch (2003) Ho (2004) e.g. ground state Energy per particle E(T = 0) = s 3EF 5 s is a Universal number

20 20 T Scale-invariance at Unitarity: a = 1 No length scale at T =0,µ=0 Quantum critical point Sachdev & Nikolic (2007) 0 1/ a =0,T =0,µ=0 Non-relativistic Conformal Field Theory Son (2008) 1/a µ AdS/CFT duality?

21 Renormalization Group: dg d` =(2 d)g g 2 2 Nikolic & Sachdev (2007) 0 Dimensionality expansions: d =2+" d =4 " Large N expansion: 2D = lower critical dimension Randeria, Duan, Shieh (1989) Expand about free fermions Expand around free bosons in Two-channel formulation Veillette, Sheehy & Radzihovsky Nikolic & Sachdev Nishida & Son (2006) 21

22 Analytical Approximations: Mean-Field + Pair Fluctuations 22 G 0 vs. G Diagrammatic Approx: Levin et al. Strinati, Perali et al. Hu, Liu & Drummond Gaussian Approx: Diener, Sensarma & Randeria Large N approx Veillette, Sheehy & Radzihovsky Nikolic & Sachdev Luttinger-Ward/Conserving Approx: Zwerger, Hausman et al. Why bother? (when there is no small parameter!) Analytical theories can give insights. E.g.: Why is reduced by 40% from its MF value?

23 Quantum Monte Carlo (QMC) Simulations: Best available tool for non-perturbative problems Fermion sign problem (sometimes absent! e.g., lattice problem with zero range attraction) Analytic continuation problem: Many types of QMC: ( or i! n!! + i0 + ) * T=0 Diffusion QMC -- wave function [Trento; Urbana; LANL, ] * Finite temperature QMC -- imaginary-time functional integrals [Amherst; Seattle; ETH; ] -- diagrammatic MC [Amherst]

24 Quantum Monte Carlo Transition Temperature Pseudogap T c <T <T unitarity T unitarity c T max c ' 0.15E F ' 0.2E F T = 0.18TF > TC (! ¹)=² F T = 0.20TF > TC T = 0.15TF = TC 1/k F a (k=k F ) 2 Burovski et al, PRL (2008) P. Magierski et al, PRL (2009, 2011)

25 Outline: (pre)history & Introduction Qualitative ideas of BCS-BEC crossover Theoretical progress Some key experiments Exact Results for strongly interacting regime Connections to other areas in physics Outlook

26 Some Key Experiments: * vortices * thermodynamics * spectroscopy * transport 26

27 Quantized Vortices in Rotating Superfluid Fermi Gases 27 6 Li Fermi gas through a Feshbach Resonance M.W. Zwierlein et al., Nature, 435, 1047, (2005)

28 Thermodynamics of the Unitary Fermi Gas [MIT; ENS; Tokyo groups] T/T F M. J. H. Ku et al., Science (2012) [Zwierlein group] λ-transition Determination of 28

29 Thermodynamics of the Unitary Fermi Gas T/T F M. J. H. Ku et al., Science (2012) [Zwierlein group] K. Van Houcke et al., Nature Phys. (2012) Expt: Zwierlein group BD QMC: UMass group 29

30 RF spectroscopy: 30 Fermion spectral function = probability to make an excitation at momentum k and energy ω k-resolved RF ~ ARPES [Jin] k-integrated RF [Grimm; Ketterle] RF threshold not at Δ but (in MF) at: QMC: Carlson & Reddy (2008) Expt: Schirotzek et al., (2008)

31 Observation of a pairing pseudogap: energy gap in the normal state near unitarity a = 1 k-resolved RF spectroscopy ß à Angle Resolved Stewart, Gaebler & Jin, Nature (2008) Photoemission (ARPES) f(!)a(k,!) Gaebler et al, Nature Phys. (2010) o unusual dispersion þ o well-defined quasiparticles? anomalous line-shape?

32 Transport coefficients: shear & bulk viscosity de dt = Z d 2 de dt = Z d 2 à Shear viscosity Dissipation in presence of a flow gradient à Bulk viscosity Dissipation with isotropic u r u 6= 0 32

33 Transport in strongly interacting fluids Shear viscosity Boltzmann equation: Sharp quasiparticles np` ` = Mean free path p` ~ ) /n ~ /s ~/k B Minimum viscosity conjecture based on AdS/CFT [Kovtun, Starinets, Son (2005)] (shear viscosity)/(entropy density) ratio of all fluids obeys: status of bound not clear, even in string theory à No known violations in the laboratory!

34 ` `min ' k 1 A digression: Mott & Ioffe-Regel à minimum conductivity conjecture Experiments à conjecture is false for charge transport! F σ T=0 Metal-Insulator transition High T incoherent transport Si:P Rosenbaum et al, PRB 27, 7509 (1983) Tyler et al, PRB 58 R (1998) 34

35 /s Experiments: Unitary Fermi gas 35 η/s KSS bound Hydrodynamic modeling of: Elliptic Flow 4 B E/E F 3 a = Damping of radial Breathing mode C. Cao, et al. Science 331,58 (2011) ~ 0.4 Data from: T. Schafer & D. Teaney, Rept.Prog.Phys. (2009) QGP (RHIC)

36 Outline: (pre)history & Introduction Qualitative ideas of BCS-BEC crossover Theoretical progress Some key experiments Exact Results for strongly interacting regime Connections to other areas in physics Outlook

37 Exact Results valid for all 1/k F a and all T/E F * Tan Relations * Sum Rules RF spectroscopy: S. Tan (2005/2008) OPE: Braaten & Platter (2008) Castin & Werner (2009) Zhang & Leggett (2009) Baym et al, PRL (2007) Punk & Zwerger, PRL (2007) Zhang & Leggett, PRA (2008) Viscosity Spectral Functions: Taylor & Randeria PRA (2010), PRL (2012) Enss, Haussmann, Zwerger, Ann. Phys.(2011) * Review: E. Braaten s chapter in Zwerger book 37

38 Tan s Contact C Two-body problem: (r) 1 r 1 a at short distances range r 0! 0 38 In the many-body problem: Density correlations hn " (r)n # (0)i C 1 r 2 1 a r k 1 F Momentum distribution: n(k) C/k 4 k k F Contact C = k 4 F F(T/E f, 1/k F a)

39 Exact results: Tan Relations Energy relation " = Z d 3 k k 2 (2 ) 3 2m n(k) C/k 4 + C 4 ma n(k) C/k 4 k k F 39 Kinetic Energy: Divergent in zero range limit Pressure relation P =2"/3+C/(12 ma) Adiabatic relation The tail that wags the dog -- E. Braaten

40 Contact & Tails of Dynamical Correlations: 40 Momentum-resolved RF A(k,!) RF Intensity lim I (!) 1!!1 4 2p m C! 3/2 Pieri, Perali & Strinati (2009) Schneider & MR (2010) Schneider & MR (2010) Dynamic Structure factor lim lim S(q,!) 2Cq 4!!1 q! m 1/2! 7/2 Son & Thomson (2010) Taylor & MR (2010)

41 Experimental studies of Contact JILA: Stewart et al, PRL 104, (2010) C/k F Momentum RF lineshape PES (k ) k and ω Tails: n(k); I(!); RF Z Fa -1 f(!)a(k,!) k-resolved RF C/k F [See also: Vale group Hoinka, PRL 110, (2013)] (k ) de da 1 S Fa -1 Thermodynamics: = h 2 4 m C. 41

42 Viscosity sum rules PRA 81, (2010) PRL 109, (2012) o Linear Response (Kubo) (!), (!) o Kramers-Kronig: causality à analyticity o Lehmann spectral representation Ed Taylor (McMaster) Valid for arbitrary interactions, all temperatures, all phases (!) o Sum rules Z Z d! (!) d! (!) = Thermodynamic Quantities! Sum rules useful for analyzing experiments constraining approximate calculations proving rigorous results 42

43 Sum-Rule, as calculated, has an ultraviolet divergence in zero-range limit =1/r 0!1 I Z d! (!) = " 3 + C a + C [Linear in 3D; log in 2D] Two-body problem has exactly the same divergence solvable limit: density n à 0, Tà 0 I 0 The difference 1 Z d! 0(!) = " C 0 a + Z 1 0 d! I C apple (!) C 0 I 0 C 0 is finite! C 15 p m! = " 3 C = Contact C 12 ma (3D) 43

44 Shear viscosity Sum Rule in 3D 1 Z 1 0 d! apple (!) C 15 p m! = " 3 C 12 ma 44 Valid for all T and all scattering lengths a P =pressure " = energy/vol. = mass density o Constraints for approximate calculations Enss, Haussmann, Zwerger, Ann. Phys. (2011) o Constraints for numerical procedures for analytic continuation of QMC data from imaginary time à real frequency Wlazłowski, Magierski, Drut, PRL (2012) o No progress on bounds yet

45 Bulk Viscosity Sum Rule in 3D 45 Valid for all T and all scattering lengths a P "/9 c 2 s At unitarity = =0 1 72ma 1 s F(T/E f, 1/k F a) But ) 2 nd law of thermodynamics for all ω and T at a = 1 Consequence of Scale invariance

46 Vanishing bulk viscosity at unitarity Bulk viscosity à relax to equilibrium after uniform dilation Scale invariance at unitarity à w.f. after scale change remains eigenstate of H à gas never leaves equilibrium under dilation (0) = 0 [Werner & Castin (2006); CFT: Son (2007)] Our result generalizes this to all frequencies 46

47 47 Measuring the shear viscosity spectral function in 3D Continuity equation ) Prediction for two-photon Bragg spectroscopy

48 Apparent scale invariance in 2D Fermi gases Expt: E. Vogt et al., PRL 108, (2012) Monopole breathing mode in 2D * Frequency! m =2! 0 * No damping Why? =0 Characteristic of scale-invariant behavior! -- without any fine tuning -- in a system with a scale: dimer binding energy " b =1/ma 2 2 E. Taylor & MR, PRL 109, (2012) Variational bound on ω b 2D Sum rule constraint on ζ Both implicate

49 Outline: (pre)history & Introduction Qualitative ideas of BCS-BEC crossover Theoretical progress Some key experiments Exact Results for strongly interacting regime Connections to other areas in physics Outlook

50 50 High Tc Superconductors Highest known Tc (in K) * cuprates Charged electrons Repulsive interactions d-wave SC doped Mott insulator competing orders: AFM, CDW, single band repulsion U >> bandwidth ξ 10 A Tc ~ ρs << Δ (underdoped) anomalous normal states - strange metal - pseudogap BCS-BEC crossover Highest known Tc/Ef ~ 0.2 * ultracold atomic gases Neutral Fermi atoms Attractive interactions s-wave SF only pairing instability single band attraction >> Ef ξ 1/kf Tc ~ ρs << Δ (for as > 0) pairing pseudogap Mean-field theory fails for Tc

51 High Tc Cuprates BCS-BEC crossover Tν d-wave normal Bose gas Tc T* Pseudo -gap s-wave Superfluid T Fermi Liquid BEC 0 BCS Superconductivity is strongest - near crossover from pair-breaking (Δ) to phase-fluctuation (ρs) dominated regimes Pseudogap when Δ > ρs Cuprate pseudogap is much more complex: Mott physics & competing orders

52 Connections with Nuclear & High Energy Physics 52 * hydrodynamic expansion of unitary Fermi gas [Thomas] ß à elliptic flow in quark-gluon plasma at RHIC ß à Viscosity/entropy bounds Ads/CFT [Son et al] * Color SC in Quark matter ß à Phases of Fermi gases with [Ketterle; Hulet] Phase separation no evidence for FFLO at unitarity [Grimm]

53 Outline: (pre)history & Introduction Qualitative ideas of BCS-BEC crossover Theoretical progress Some key experiments Exact Results for strongly interacting regime Connections to other areas in physics Outlook

54 BCS-BEC Crossover: Experiments & Theory A remarkable new chapter in many-body physics Unitary Fermi gas: a new paradigm for strongly interacting systems Open questions -- Is there a general upper bound on Tc/Ef? -- 2D systems -- vortices, solitons, -- transport -- non-equilibrium dynamics -- non-s-wave, SOC & topological states 54

55 The End! 55